Results 1  10
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143
Spectral Theory Of Elliptic Operators On NonCompact Manifolds
, 1992
"... preliminaries Let H be a complex Hilbert space, A a densely dened linear operator in H (the domain of A will be denoted D(A)). Suppose that A has a closure A or, equivalently, that the adjoint operator A is densely dened (see e.g. [32]). We shall denote by GA the graph of A i.e. the set of pairs ..."
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Cited by 61 (9 self)
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preliminaries Let H be a complex Hilbert space, A a densely dened linear operator in H (the domain of A will be denoted D(A)). Suppose that A has a closure A or, equivalently, that the adjoint operator A is densely dened (see e.g. [32]). We shall denote by GA the graph of A i.e. the set of pairs fu; Aug; u 2 D(A). Then G A = GA , i.e. the graph of A is the closure of the graph of A. Moreover A = A = (A ) . Now let A + be another densely dened linear operator in H. DEFINITION 1.1. A + is called formally adjoint to A if (1:1) (Au; v) = (u; A + v); u 2 D(A); v 2 D(A + ); where (; ) is the scalar product in H. If A = A + then A is called symmetric or formally self{adjoint. Note that since A; A + are densely dened, both A and A + have closures. DEFINITION 1.2. Let A; A + be as in Denition 1.1. Then the minimal and the maximal operator for A are dened as follows: A min = A = A ; A max = (A + ) : Note that both A min and A max are...
Uniformly elliptic operators on Riemannian manifolds
 J. Diff. Geom
, 1992
"... Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiiso ..."
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Cited by 32 (2 self)
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Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiisometric to g. We first prove some Poincare and Sobolev inequalities on geodesic balls. Then we use Moser's iteration to obtain Harnack inequalities. Gaussian estimates, uniqueness theorems, and other applications are also discussed. These results involve local or global lower bound hypotheses on the Ricci curvature of g. Some of them are new even when applied to the Laplace operator of (M, g). 1.
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
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Cited by 26 (3 self)
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We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
Manifolds and Graphs With Slow Heat Kernel Decay
 Invent. Math
, 1999
"... We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal. ..."
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Cited by 24 (2 self)
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We give upper estimates on the long time behaviour of the heat kernel on a noncompact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal.
Uniqueness of the Ricci flow on complete noncompact manifolds
, 2005
"... The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80’s, Shi [20] generalized the local existence result t ..."
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Cited by 24 (5 self)
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The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80’s, Shi [20] generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ricci flow with surgery. In this paper, we give an affirmative answer for the uniqueness question. More precisely, we prove that the solution of the Ricci flow with bounded curvature on a complete noncompact manifold is unique.
Moduli spaces of critical Riemannian metrics in dimension four
"... Abstract. We obtain a compactness result for various classes of Riemannian metrics in dimension 4; in particular our method applies to antiselfdual metrics, Kähler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can ..."
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Cited by 23 (0 self)
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Abstract. We obtain a compactness result for various classes of Riemannian metrics in dimension 4; in particular our method applies to antiselfdual metrics, Kähler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can be compactified by adding metrics with orbifold singularities. Similar results were obtained for Einstein metrics in [And89], [BKN89], [Tia90], but our analysis differs substantially from the Einstein case in that we do not assume any pointwise Ricci curvature bound. 1.
Gaussian Upper Bounds For The Heat Kernel On Arbitrary Manifolds
, 1997
"... In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the conventional ..."
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Cited by 22 (2 self)
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In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the conventional Laplace operator in R...
Riesz transform, Gaussian bounds and the method of wave equation
 Math. Z
"... Abstract. For an abstract selfadjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We al ..."
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Cited by 20 (1 self)
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Abstract. For an abstract selfadjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on L p for all p ∈ (1, 2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L p of the LaplaceBeltrami operator on Riemannian manifolds for p> 2. 1.